If a²/(b² + c²) = b²/(c² + a²) = c²/(a² + b²), then what is the value of a⁴ + b⁴ + c⁴ equal to?

Let each ratio equal k. Then a² = k(b² + c²), b² = k(c² + a²), c² = k(a² + b²). Adding: a² + b² + c² = 2k(a² + b² + c²), so k = 1/2. Therefore 2a² = b² + c², 2b² = c² + a², 2c² = a² + b². From these we can derive a² = b² = c² only if the system is consistent, but more directly: squaring and using these relations gives (a² + b² + c²)² = a⁴+b⁴+c⁴ + 2(a²b²+b²c²+c²a²). Also a² + b² + c² = 2k(sum) implies... actually the relations 2a² = b²+c² etc. imply a² = b² = c², so a⁴+b⁴+c⁴ = 3a⁴ and a²b²+b²c²+c²a² = 3a⁴, giving the ratio 1. Answer: a²b² + b²c² + c²a².